Intravalley scattering

Intravalley scattering refers to processes where the initial and final valley of the scattered electron is the same. The scattering can take place through acoustic or optical phonons.

Acoustic phonons

The scattering by acoustic phonons in the same valley is assumed to be elastic by the equipartition approximation. The total scattering rate is given by

            √ --          2√ --
Pe-,ac(E) = --2mDOSkBT--Ξ--  E;                                                 (A.1)
               π ℏ4u2ρD
        u = 1-(2u  + u );                                                        (A.2)
            3    t   l
   m      = (m m2 )13 .                                                          (A.3)
     DOS       l t
The meaning and value of the parameters is given in Table A.1.


Table A.1: Parameters for intravalley acoustic phonon scattering
SymbolMeaning Value
mDOS effective mass for density of states
ml effective mass in longitudinal direction 0.98mo
mt effective mass in transverse direction 0.19mo
m0 mass of an electron 9.11 × 10-31kg
ρD material (silicon) density 2338.0kg m-3
u average sound (acoustic phonon) velocity
ul sound velocity in transverse direction 5.410 × 107m s-1
ut sound velocity in longitudinal direction 9.033 × 107m s-1
Ξ scalar representing average of potential deformation tensor 7.2eV

Optical phonons

Intravalley scattering by optical phonons is inelastic; energy is lost (gained) through the emission (absorption) of an optical phonon.

The total scattering rate as a function of energy is given by

                         [         ]
            (DtK--)2mDOS--    Nop    ∘  ---------
Pe-,op(E ) = √2-πℏ3ρ  ω     Nop + 1     E ± ℏωop,                                        (A.4)
                   D  op
where the top branch corresponds to the rate for absorption and the bottom branch corresponds to the rate for emission.

The value Nop denotes the mean occupancy number for optical phonons under equilibrium conditions and is given by a Bose-Einstein distribution:

      ( ℏω      ) -1
Nop =   ---op-- 1    .                                                        (A.5)
        kBT
The meaning and value of the other newly-introduced parameters are given in Table A.2.

Intravalley scattering with optical phonons is forbidden in the X- and Γ-valleys of silicon, due to symmetry considerations; only the L-valley electrons can undergo intravalley scattering in silicon [164].


Table A.2: Parameters for intravalley optical phonon scattering
SymbolMeaning Value
DtK optical coupling constant 2.2 × 1010eV m-1
ωop radial frequency of optical phonon
ωop optical phonon energy 0.0612eV